Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Two riddles

2 marks

1.1. You have three coins, which appear identical, except one of them is counterfeit and does not have the same weight as the other two. You have a set of balance scales, so you can compare coins against each other.

Can you find the fake coin?

No, it’s impossible.

Not enough information to decide.

Yes, you can always find the fake coin in one use of the scales.

Yes, you can always find the fake coin in one or two uses of the scales.

Yes, you can always find the fake coin, but it will take three uses of the scales.

You would start by placing one coin (A & B) on each side of the scales. If they balance, then you know that the other coin C is the fake.

If, however, they are unequal then then you don’t know if A is the fake or B is the fake (but you know C is genuine). Next, compare A & C – if they balance, then B must the fake. If they are unequal, then we know A is fake, because it matches neither B or C.

So the answer is “Yes, you can always find the fake coin in one or two uses of the scales.”

2 marks

1.2. A builder agrees to fix your house in 7 days, and you agree to pay her a bar of gold. However, she wants to receive 1/7 of her payment at the end of each day, so you will need to cut the bar of gold into pieces. Of course, you could cut the gold into 7 pieces and hand over one piece at the end of each day, but you only want to cut the gold bar into three pieces. What three pieces would enable you to make the correct daily payments.

It’s impossible.

1/7, 1/7, 1/7

1/7, 1/7, 5/7

2/7, 2/7, 3/7

1/7, 2/7, 4/7

1/7, 2/7, 4/7 works because you can do the following:

2. Exoplanets

One of the most exciting new areas of astronomical research in my lifetime has been the hunt for exoplanets. The term describes planets that are not in our solar system, but rather they orbit distant stars. This short video explains how you discover an exoplanet.

1 mark

2.1 Some exoplanets exist in the Goldilocks Zone. Which of these statements does not apply to planets in this zone?

They are not too close to their star and not too far away.

They have a mass that is just about right.

They are more likely to sustain life than other planets.

They are not too hot and not to cold.

They are more likely to have water than other planets.

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1 Weighing the baby at the clinic was a problem. The baby would not keep still and caused the scales to wobble. So I held the baby and stood on the scales while the nurse read off 78 kg. Then the nurse held the baby while I read off 69 kg. Finally I held the nurse while the baby read off 137 kg. What was the combined weight of all three?

142 kg

147 kg

206 kg

215 kg

284 kg

We let the weights of the baby, the nurse and myself be x kg, y kg and z kg, respectively. The information we are given implies that x+z=78, x+y=69 and y+z=137.

Adding these three equations gives x+z+x+y+y+z=78+69+137, that is, 2x+2y+2z=284.

It follows x+y+z=142. So the combined weight of all three was 142 kg.

3 marks

3.2 What was the baby’s weight?

Correct Solution: 5 kg

From the previous answer, we know all three (me + nurse + baby) weigh 142 Kg.

And we already know from the question that (me + nurse) weigh 137 Kg, so the baby must weigh 5Kg.

Alternatively, there is a longer more methodical approached, outlined below.